Question 159
Let $n$ be a positive integer. Ana and Banana play a game. Banana thinks of a function $f: \mathbb{Z} \to \mathbb{Z}$ and a prime number $p$. He tells Ana that $f$ is nonconstant, $p \lt 100$, and $f(x+p) = f(x)$ for all integers $x$. Ana's goal is to determine the value of $p$. She writes down $n$ integers $x_1, \dots, x_n$. After seeing this list, Banana writes down $f(x_1), \dots, f(x_n)$ in order. Ana wins if she can determine the value of $p$ from this information. Find the smallest value of $n$ for which Ana has a winning strategy.